Generating artificial intelligence plans of high diversity

ABSTRACT

In an approach for improved artificial intelligence planning in an automated machine learning pipeline, a processor formulates an artificial intelligence planning problem. A processor receives a pre-defined stopping criterion for generating one or more plans for the artificial intelligence planning problem. A processor generates the one or more plans by executing a planning algorithm. A processor reformulates the artificial intelligence planning problem into a new artificial intelligence planning problem by forbidding plans that correspond to super-sets of the one or more plans. A processor generates one or more new plans based on the reformulation until the pre-defined stopping criterion is reached.

BACKGROUND

The present disclosure relates generally to the field of artificialintelligence, and more particularly to automated planning.

Automated planning is a long-standing sub-area of artificialintelligence that aims at solving problems that involve finding astrategy of action, provided that the problems are modeled in a suitableinput language. Automated planning is a branch of artificialintelligence that concerns the realization of strategies or actionsequences, typically for execution by intelligent agents, autonomousrobots and unmanned vehicles. Unlike classical control andclassification problems, the solutions are complex and must bediscovered and optimized in multidimensional space. Planning is alsorelated to decision theory. In known environments with available models,planning can be done offline. Solutions can be found and evaluated priorto execution. In dynamically unknown environments, the strategy oftenneeds to be revised online. Models and policies must be adapted.Solutions usually resort to iterative trial and error processes commonlyseen in artificial intelligence. These include dynamic programming,reinforcement learning and combinatorial optimization. Languages used todescribe planning and scheduling are often called action languages.

Given a description of the possible initial states of the world, adescription of the desired goals, and a description of a set of possibleactions, the planning problem is to synthesize a plan that is guaranteed(when applied to any of the initial states) to generate a state whichcontains the desired goals (such a state is called a goal state). Thedifficulty of planning is dependent on the simplifying assumptionsemployed. Several classes of planning problems can be identifieddepending on the properties the problems have in several dimensions.

SUMMARY

Aspects of an embodiment of the present disclosure disclose an approachfor improved artificial intelligence planning in an automated machinelearning pipeline. A processor formulates an artificial intelligenceplanning problem. A processor receives a pre-defined stopping criterionfor generating one or more plans for the artificial intelligenceplanning problem. A processor generates the one or more plans byexecuting a planning algorithm. A processor reformulates the artificialintelligence planning problem into a new artificial intelligenceplanning problem by forbidding plans that correspond to super-sets ofthe one or more plans. A processor generates one or more new plans basedon the reformulation until the pre-defined stopping criterion isreached.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram illustrating an automatingartificial intelligence pipeline generation environment, in accordancewith an embodiment of the present disclosure.

FIG. 2 is a flowchart depicting operational steps of a plan generatingmodule within a computing device of FIG. 1 , in accordance with anembodiment of the present disclosure.

FIG. 3 illustrates an exemplary functional diagram of the plangenerating module within the computing device of FIG. 1 , in accordancewith an embodiment of the present disclosure.

FIG. 4 illustrates exemplary operational steps of the plan generatingmodule of FIG. 1 , in accordance with an embodiment of the presentdisclosure.

FIG. 5 illustrates an exemplary experimental evaluation of the plangenerating module within the computing device of FIG. 1 , in accordancewith an embodiment of the present disclosure.

FIG. 6 illustrates another exemplary experimental evaluation of the plangenerating module within the computing device of FIG. 1 , in accordancewith an embodiment of the present disclosure.

FIG. 7 is a block diagram of components of the computing device of FIG.1 , in accordance with an embodiment of the present disclosure.

DETAILED DESCRIPTION

The present disclosure is directed to systems and methods for improvedartificial intelligence planning in an automated machine learningpipeline.

Embodiments of the present disclosure recognize a need for automatedplanning that aims at solving problems that involve finding a course ofaction (aka plan) provided the problems are modeled in a suitable inputlanguage. Embodiments of the present disclosure disclose exploring thespace of valid plans, with a focus on diversity of the set of plans, aswell as the quality of each individual plan. Embodiments of the presentdisclosure disclose super-(multi)set top quality planning which skipplans that are supersets of previously found plans. Embodiments of thepresent disclosure disclose super-(multi)set diverse planning which skipplans that are supersets of previously found plans. In an example, intop quality/diverse planning, many plans are effectively equivalent fromthe application perspective. For instance, an order in which someactions are performed can make no difference, or additional unnecessaryactions can be part of a valid plan. While the unordered top-qualityplanning may alleviate this problem of indifference regarding actionordering, the unordered top-quality planning may still capture all planswith unnecessary actions, and thus make the set of valid plansprohibitively large and sometimes (unnecessarily) infinite.

Embodiments of the present disclosure disclose exploring the space ofvalid plans by iteratively finding a plan and reformulating the planningproblem to forbid a set of plans. Embodiments of the present disclosuredisclose reformulations that forbid plans that, when viewed as(multi)sets of actions, are super(multi)sets of given (multi)sets ofactions. These reformulations may work for both the top-quality anddiverse settings. Embodiments of the present disclosure disclose systemsand methods of improved artificial intelligence planning in an automatedmachine learning pipeline. Embodiments of the present disclosuredisclose receiving an artificial intelligence planning problem andreceiving a stopping criterion, the stopping criterion associated withthe number of artificial intelligence plans needed or a quality ofartificial intelligence plans needed. Embodiments of the presentdisclosure disclose utilizing a cost-optimal planner to generate anupdated solution to the planning problem based on the stoppingcriterion.

Embodiments of the present disclosure disclose computing a set of topquality or diverse plans, extending the criteria of equivalence fromset/multiset equivalence to super-(multi)sets. For example, a plan pwould be considered interesting with regarding to an existing set ofplans P if p (viewed as a multiset of operators) is not a super-multisetof any plan in P. Embodiments of the present disclosure discloseiterative methods that find a plan, forbid the plan by reformulating theproblem and find the next plan by forbidding exactly a plan provided andall its possible super-multisets, or forbidding exactly a collection ofplans provided and all their possible super-multisets.

Embodiments of the present disclosure disclose formulating a finitedomain representation planning problem (or multiple planning problems).Embodiments of the present disclosure disclose receiving a stoppingcriterion of either a quality bound or a number of plans or both.Embodiments of the present disclosure disclose running a cost-optimalartificial intelligence to obtain one plan. Embodiments of the presentdisclosure disclose reformulating a finite domain representationplanning problem into a new finite domain representation planningproblem so that exactly the found plans and all plans that correspond tosuper-sets of those plans are forbidden in the iterations. For example,variables may be extended to track the operators on the previously foundplans and operator sets checked. Operators may be modified and extendedto change the values of extra variables. An initial state may beextended to allocate initial values to the extra variables. A goal maybe extended to allocate goal value to the operator sets checkingvariable.

Embodiments of the present disclosure disclose running astate-of-the-art planner until a stopping criterion is reached: either aplan of worse quality than the quality bound is found, or enough planswere found, or there are no more plans exist. Embodiments of the presentdisclosure disclose computing a set of diverse plans, where thecost-optimal planner may be replaced by an agile/satisficing planner,that provides a solution of lower cost, but much faster. Embodiments ofthe present disclosure disclose that the reformulation forbids insteadall the found plans and all plans that correspond to super-multisets ofthose plans. Embodiments of the present disclosure disclose directlyenabling a new, improved, scenario planning process. Embodiments of thepresent disclosure disclose quickly generating plans for multiplemodernization workflows. Embodiments of the present disclosure disclosequickly skipping many irrelevant machine learning pipelines.

Embodiments of the present disclosure disclose top-quality planning ingeneral and quotient top-quality planning in particular deal withproducing multiple high-quality plans while allowing for their efficientgeneration, skipping equivalent ones. Embodiments of the presentdisclosure disclose considering a different relation: two plans arerelated if one's operator multiset is a subset of the other's.Embodiments of the present disclosure disclose novel reformulations thatforbid plans that are related to the given ones. Embodiments of thepresent disclosure disclose defining a new subset top-quality planningproblem, with finite size solution sets, while the new relation is nottransitive and thus not an equivalence relation.

Embodiments of the present disclosure disclose a computational problemunder the umbrella of top-quality planning that allows to furtherrestrict the solution set. Embodiments of the present disclosuredisclose considering the following relation: a plan is related toanother if its operator multiset is a subset of the other's. Embodimentsof the present disclosure disclose a planning task reformulation thatforbids plans whose operator multisets are supersets of the given ones.Embodiments of the present disclosure disclose the solution sets of thenew subset top-quality planning problem are of finite size. Embodimentsof the present disclosure disclose iteratively finding and forbiddingplans, exploiting the new reformulation. Embodiments of the presentdisclosure disclose a reformulation that forbids plans that aresupersets as sets rather than multisets, depicted by top-SQ.

Embodiments of the present disclosure disclose a reformulation that,given a set of plans, forbid all plans that are supersets (as operatormultisets) of some plan in the given set. In other words, thereformulation may preserve all plans that are not supersets of all theplans in the given set. That allows for both ignoring plan reorderingsand unnecessary operators. Embodiments of the present disclosuredisclose the reformulation keeps track of the original operators fromthe planning task applied on the way to the goal with the extra effectscounting the number of applications of each operator applied to reachthe goal. Once the goal is reached, if there is an operator with thenumber of applications lower than the number of appearances in amultiset, then the found plan is not a superset of that multiset. Thereformulation may then switch to the second phase in which it ensuresthat there exists at least one operator from the multiset, whosecorresponding operator was not applied and now can be applied.

The present disclosure will now be described in detail with reference tothe Figures. FIG. 1 is a functional block diagram illustrating anautomating artificial intelligence pipeline generation environment,generally designated 100, in accordance with an embodiment of thepresent disclosure.

In the depicted embodiment, automating artificial intelligence pipelinegeneration environment 100 includes computing device 102 and network108.

In various embodiments of the present disclosure, computing device 102can be a laptop computer, a tablet computer, a netbook computer, apersonal computer (PC), a desktop computer, a mobile phone, asmartphone, a smart watch, a wearable computing device, a personaldigital assistant (PDA), or a server. In another embodiment, computingdevice 102 represents a computing system utilizing clustered computersand components to act as a single pool of seamless resources. In otherembodiments, computing device 102 may represent a server computingsystem utilizing multiple computers as a server system, such as in acloud computing environment. In general, computing device 102 can be anycomputing device or a combination of devices with access to plangenerating module 110 and network 108 and is capable of processingprogram instructions and executing plan generating module 110, inaccordance with an embodiment of the present disclosure. Computingdevice 102 may include internal and external hardware components, asdepicted and described in further detail with respect to FIG. 7 .

Further, in the depicted embodiment, computing device 102 includes plangenerating module 110. In the depicted embodiment, plan generatingmodule 110 is located on computing device 102. However, in otherembodiments, plan generating module 110 may be located externally andaccessed through a communication network such as network 108. Thecommunication network can be, for example, a local area network (LAN), awide area network (WAN) such as the Internet, or a combination of thetwo, and may include wired, wireless, fiber optic or any otherconnection known in the art. In general, the communication network canbe any combination of connections and protocols that will supportcommunications between computing device 102 and plan generating module110, in accordance with a desired embodiment of the disclosure.

In one or more embodiments, plan generating module 110 is configured toformulate an artificial intelligence planning problem. Plan generatingmodule 110 may receive the artificial intelligence planning problem forvarious applications, e.g., a risk management application, an energydomain application, a healthcare application, and a malware detectionapplication. Plan generating module 110 may compute a set of top-qualityplans via artificial intelligence planning algorithms. For example, plangenerating module 110 may find a strategy of action, provided that theplanning problem is modeled in a suitable input language. Plangenerating module 110 may find the one best solution to a problem, forexample, in a top K planning seeking to find the K best solutions. Forexample, given quality bound q, the top-quality planning is to findplans with quality under the bound. An unordered top-quality planning isto skip plans that are (multi)set-equivalent to previously found ones. Adiverse planning is to find k plans of a particular requirement ondiversity (e.g., variety of computational problems), given a bound k onthe number of plans and a diversity metric D. In an example, plangenerating module 110 may consider two plans are related if one'soperator multiset is a subset of the other's. Plan generating module 110may provide reformulations that forbid plans that are related to thegiven ones. While the new relation may not be transitive and thus not anequivalence relation, plan generating module 110 may define a new subsettop-quality planning problem, with finite size solution sets.

In one or more embodiments, plan generating module 110 is configured toreceive a pre-defined stopping criterion for generating one or moreplans for the artificial intelligence planning problem. The pre-definedstopping criterion may be associated with one or more of a number ofartificial intelligence plans needed or a quality of artificialintelligence plans needed. For example, the pre-defined stoppingcriterion can be a stopping criterion of either a quality bound or anumber of plans or both. Top-quality planning may deal with generatingall high-quality plans up to a certain bound. The need for producingsuch a collection of plans may be established for many applications,including plan recognition, business process automation, and automatedmachine learning. Top-quality planning may serve as a basis for solvingother computational problems, such as quality-aware diverse planning. Inthese and other applications, the choices of planning models, whetheravoidable or not, may have unintended effects on plans. These include(zero-cost) loops, unnecessary repeated operator applications,continuing after the goal is reached. While producing all high-qualityplans can be challenging, and even impossible when there are infinitelymany such plans, one practical approach suggested to generate all plansthat are not equivalent when ignoring operator ordering.

In one or more embodiments, plan generating module 110 is configured togenerate one or more plans by executing a planning algorithm (e.g., anartificial intelligence planner). In an example, a planning algorithmcan be a cost-optimal planner, an agile planner, a satisficing planner,a top-k planner, a top-quality planner, a diverse planner or any othersuitable planner. Plan generating module 110 may obtain a plan byexecuting the planning algorithm. Plan generating module 110 may utilizea cost-optimal artificial intelligence planner to generate an updatedsolution to an artificial intelligence planning problem based on apre-defined stopping criterion. Plan generating module 110 may run acost-optimal artificial intelligence planner to obtain one plan. Inanother example, plan generating module 110 may run a top-k planner, atop-quality planner, a diverse planner, an agile planner, a satisficingplanner, or any other planner to generate a solution to a planningproblem. Plan generating module 110 may compute a set of top-qualityplans using a post-process plan by filtering out equivalent orirrelevant plans. Plan generating module 110 may translate the remainingplans into a solution for an application. Plan generating module 110 maycompute a set of top quality or diverse plans, extending the criteria ofequivalence from set/multiset equivalence to super-(multi)sets.

In one or more embodiments, plan generating module 110 is configured toreformulate an artificial intelligence planning problem into a newartificial intelligence planning problem to forbid plans that correspondto a super set of the plan. Plan generating module 110 may reformulate afinite domain representation planning problem into a new finite domainrepresentation planning problem so that exactly the found plans and allplans that correspond to super-sets of those plans are forbidden in aniteration. For example, plan generating module 110 may extend variablesto track operators on the previously found plans and operator setschecked. Plan generating module 110 may modify and extend the operatorsto change the values of extra variables. Plan generating module 110 mayextend an initial state to allocate an initial value to extra variables.Plan generating module 110 may extend a goal to allocate goal value tothe operator sets checking variable. Plan generating module 110 mayforbid all the found plans and all plans that correspond tosuper-multisets of those plans. Plan generating module 110 may compute aset of top quality or diverse plans, extending the criteria ofequivalence from set/multiset equivalence to super-(multi)sets. Plangenerating module 110 may find a plan, forbid the plan by reformulatingthe problem, and find the next plan. Plan generating module 110 mayforbid exactly a plan provided and all the plan's possiblesuper-multisets. Plan generating module 110 may forbidding exactly acollection of plans provided and all their possible super-multisets.

Forbidding Plans as Super-Multisets

In an example, a computational problem of interest as well as a methodfor solving the problem can be defined as follows. First, letR_(⊂)={(π,π′)|MS(π)⊂MS(π′)} denote the relation defined by the subsetoperation over plan operator multisets.

Definition 1 (Subset Top-Quality Planning Problem)

Given a planning task Π and a natural number q, find a set of plansP⊂P_(πs.t.)(i). ∀_(π)∈P, cos(π)≤q , and (ii) ∀_(π),∈P_(Π)\P with costwith cos(π)≤q , ∃_(π)∈Ps.t.(π,π′)∈R_(⊂).

In words, a plan π with cos(π)≤q may not be part of the solution to thesubset top-quality planning problem only if its subset is part of thesolution. Note, while a top quality and unordered top-quality solutionsare also subset top-quality solutions, plan generating module 110 mayfind smallest (in the number of plans) such solutions. While unorderedtop-quality solutions can be of infinite size, smallest subsettop-quality planning solutions are always finite.

Theorem 1 Given a planning task and a natural number q, a smallestsubset top-quality planning problem solution P is of finite size.

Proof: For a plan π that contains a cycle, if π′ is obtained from π byremoving all cycles, then we have (π,π′)∈R_(⊂). Thus, P is a subset ofthe set of all cycle-free plans. Since H has only a finite number ofstates, the number of cycle-free plans is also finite.

Plan generating module 110 may find smallest subset top quality planningproblem solutions. Following the line of work on reformulating planningtasks to forbid a set of plans, plan generating module 110 may present areformulation that, given a set of plans, forbid all plans that aresupersets (as operator multisets) of some plan in the given set. Inother words, the reformulation preserves all plans that are notsupersets of all the plans in the given set. That allows for bothignoring plan reorderings and unnecessary operators. Plan generatingmodule 110 may forbid multiple plans as operator super-multi-sets(FOSMS).

Definition 2 Let Π=(V, O, s_(o), s_(*), cost) be a planning task, M={M₁,. . . , M_(k)} be a set of operator multisets, and U⊆O be the set of alloperators in the multisets in M. The task (Π_(M) ⁻=

V′, O′, s′_(o), s′_(*), cost′) is defined as follows.

V′=V∪{v}∪{v _(o)|o∈U}, where the variable v has the domain dom(v={0 . .. k} and dom(v _(o))={0, . . . , m(o)}, where m(o)=max_(i=1)^(k){m_(Mi)(o)},

O′={ o |o∈O, 0≤i≤ m (o)}∪{o _(i,j) |o∈M _(j), 1≤j≤k, 0≤i<m _(Mi)(o)},where

pre( o _(i))=pre(o)∪{

v ,0

}∪{

v _(o) i

|o∈U},

eff( o _(i))=eff(o)∪{

v _(o),i+1

o∈U,i< m (o)},

o _(i,j)=

s _(*)∪{

v _(o) ,i

,

v ,j−1},{ v ,j

}

and

cost′( o _(i))=cost(o),cost′(o _(i,j))=0,

s′ ₀ =s ₀∪{

v ,0

}∪{

v _(o),0

|o∈U}, and

s′ _(*) =s _(*)∪{

v ,k

}.

In words, the reformulation keeps track of the original operators fromthe planning task H applied on the way to the goal (i.e., ō_(i)) withthe extra effects counting the number of applications of each operatorapplied to reach the goal. Once the goal is reached, if there is anoperator with the number of applications lower than the number ofappearances in a multiset, then the found plan is not a superset of thatmultiset. The reformulation then switches to the second phase in whichit ensures that there exists at least one operator o from the multisetM_(j), whose corresponding operator ō_(i) was not applied and nowo_(i,j) can be applied. Note, the goal can only be reached if

v, k

is true in addition to the original goal s_(*). That is the goal can bereached by applying the additional operators o_(i,j), which areapplicable only when the original goal was achieved and thecorresponding original operator was not applied m_(Mi)(o) times.

Algorithm 1: Iterative subset top-quality planning.

Input: Planning task Π, quality bound q

-   -   P→Φ, Π′→Π, π→shortest cost-optimal plan of Π′    -   while cost(π)≤q do    -   π→MAPPLANBACK(π)    -   P→P∪π∪{π′|π′|π, MS(π′)≠MS(π)}    -   Π′→Π_(M+) ⁻, where M=MS(π′)|π′∈P}    -   π→shortest cost-optimal plan to Π′    -   return P

Consider an example task Π with O={a, b, c, d}, plans aba and ca andtheir corresponding multisets M={{a, a, b}, {a, c}}. The reformulatedtask Π_(M+) ⁻ has extra variables: ternary v and v _(a), as well asbinary v _(b) and v _(c). O′ includes multiple copies of the originaloperators that appear in U. Here, we have O′=ā₀, ā₁, ā2, b ₀, b ₁, c ₀,c ₁, d ₀, ∪{a_(0,1), a_(0,2), a_(1,1), a_(1,2), b_(0,1), c_(0,2)}.Consider a plan π=abd for Π and a sequence of operators π=ā₀ b ₀ d₀a_(1,1)c_(0,2). Note that π is a plan for Π_(M+) ⁻: ā₀ b ₀ d ₀ achievesthe original goal, while a_(1,1) and c_(0,2) are changing the value of vfrom 0 to 1 and then to 2. Thus, π is not forbidden.

To solve the subset top-quality planning problem, one can exploit thereformulation FOSMS in Definition 2, find a plan, extend the plan to aset of plans with the help of structural symmetries, reformulate theoriginal planning task using the set of plans found so far, extend theset by solving the reformulated task optimally, adding the cost optimalplan (and possibly its symmetric plans to the set), continue until thereare no more plans of cost lower or equal to q. The detailed scheme isdepicted in Algorithm 1.

Theorem 2 Algorithm 1 is sound and complete for subset top-qualityplanning when using cost-optimal planners that find shortestcost-optimal plans.

Proof: First, according to Theorem 1, Algorithm 1 will terminate andreturn a set of plans. Let P={π₁, . . . , π_(k)} be plans for Π withtheir corresponding multisets M={MS(π₁), . . . , MS(π_(k))}. Let πt=a¹ .. . a^(n) be some plan for Π such that MS (π_(i))⊂/MS(π) for 1≤i≤k. Itis shown that there exists a corresponding to π plan π for Π_(M+) ⁻. Letπ ₁=a_(l) ₁ ¹ . . . a_(l) _(n) ^(n) be the applicable sequence ofoperators from O′ that corresponds to π=a¹ . . . a^(n) and s₁ be the endstate of applying π₁ in s′₁. Then, s_(*)⊆s₁ and s₁[v]=0. SinceMS(π_(i))⊂/MS(π), there exists an operator b^(i)∈MS(π_(i)) such thatb^(i)∉MS(π). Thus, we have s₁[v _(b) _(i) ]=0,1≤i≤k. Then, π ₂=b_(0,1) ¹. . . b_(0,k) ^(k) is applicable in s₁ and results in a state s₂ suchthat s_(*)⊆s₂ and s₂[v]=k. In other words, π=π ₁ π ₂ is a plan forΠ_(M+) ⁻.

The relation R∪ is reflexive, antisymmetric, and transitive and thusdefines a partial order over plans. Let π and π′ be two plans such that(π, π′)∈R∪. Then cost(π)≤cost(π′) and the plan π is strictly shorterthan π′ and therefore a planner that finds shortest among the optimalplans will find π before π′. Thus, Algorithm 1 finds plans that areminimal under the partial order defined by R∪ and will terminate when itfinds all such plans of cost less or equal q.

Forbidding Plans as Super-Sets

The reformulation in Definition 2 might in some cases be unnecessarilypermissive, allowing plans that are supersets as sets but not asmultisets. The stricter reformulation will, therefore, given a set ofplans, forbid all plans that are supersets (as operator sets) of someplan in the given set. This may be called as forbidding multiple plansas operator super-sets (FOSS).

Definition 3 Let Π=(V, O, s_(o), s_(*), cost) be a planning task, X={X₁,. . . , X_(k)} be a set of operator sets, and U=U_(i=1) ^(k)X_(i)⊂O bethe union of these sets. The task Π_(X+) ⁻==

V′, 0′, s′_(o), s′_(*),cost′

is defined as follows.

V′=V∪{v}∪{v _(o)|o∈U}, where the variable v has the domain dom(v)={0 . .. k} and all additional variables v _(o) being binary variables,

O′={ō _(i) |o∈O}∪{o _(i) |∈X _(j), 1≤j≤k}, where

ō=

pre(o)∪{

v,0

}, eff(o)∪{

v _(o),1

|o⊂U},

o _(i) =

s _(*) ∪{

v _(o),0

,

v,i−1

},{

v,i

}

, and

cost′(ō _(i))=cost(o),cost′(o _(i,j))=0,

s′ ₀ =s ₀ ∪{

v,0

}∪{

v _(o),0

|o∈U}, and

s′ _(*) =s _(*)∪{

v,k

}.

X_(i), 1≤i≤k are the operator sets associated with the plans seen so farfor the original planning task H. In words, the reformulation keepstrack of the original operators from the planning task applied on theway to the goal (i.e., ō) with the extra effect (v _(o),1) for theoperators in the set U, and then switches to the second phase, in whichit ensures that there exists at least one operator from the originalplans, that was not applied and now can be applied. Note, k is the sizeof the set X, and the goal can only be reached if (v, k) is true inaddition to the original goal, s,, . Thus, the goal can be reached byapplying the additional operators o_(i), which are applicable only whenthe original goal was achieved and the corresponding original operatorwas not applied. Consider the same example planning task Π overoperators O={a, b, c, d}, the same two plans aba and ca and theircorresponding sets X={{a, b}, {a, c}. The reformulated task Π_(X+) ⁻would then have extra ternary variable i and binary variables v _(a), v_(b) and v _(c). O′ includes multiple copies of the original operatorsthat appear in U. Here, there is O′={ā, b, c, d}∪{a₁, a₂, b₁c₂}.Consider again the plan π=abd for Π and a sequence of operators π=ābd.Applying π to the initial state of Π_(X+) ⁻ results in a states′_(*)=s_(*)∪{

v,0

,

v _(a),1

,

v _(b),1

,

v _(c),0

,

v _(d),1

}. Note that none of {a₁, a₂, b₁c₂} operators are applicable in s′ andtherefore there is no corresponding to π plan of Π_(x+) ⁻. Observe thatthe reformulation, while forbidding plans that are supersets of elementsin X, allows plans that are proper subsets of these sets in X. Forexample, a plan abc is a superset (as operator set) of aba. Assuming thereformulation is used within Algorithm 1, if aba is found first, abc isforbidden, but not vice versa. Thus, Algorithm 1 can be slightly adaptedto check whether the newly found plans are subsets of previously foundones, discarding the plans that are supersets. Note that this could nothappen in the case of Definition 2, since a plan that corresponds to aproper subset (as multiset) of another one will be found earlier.

It is worth noting here that in some cases FOSS might be preferred overFOSMS or vice versa. One example of such a case is whether it isimportant to differentiate alternative but similar ways of achievingsubgoals.

In one or more embodiments, plan generating module 110 is configured togenerate one or more new plans based on the reformulation until thepre-defined stopping criterion is reached. Plan generating module 110may run until a stopping criterion is reached: either a plan of worsequality than the quality bound is found, or enough plans were found, orthere are no more plans exist. In an example, plan generating module 110may generate a subset top- quality solution in domain with infinite sizeunordered top-quality solutions. Plan generating module 110 may utilizea cost-optimal planner to generate an updated solution to the artificialintelligence planning problem based on the pre-defined stoppingcriterion. Plan generating module 110 may solve a computational problemunder the umbrella of top-quality planning that allows to furtherrestrict the solution set. Plan generating module 110 may consider thefollowing relation: a plan is related to another if its operatormultiset is a subset of the other's. Plan generating module 110 mayprovide a planning task reformulation that forbids plans whose operatormultisets are supersets of the given ones. Plan generating module 110may prove that the solution sets of the new subset top-quality planningproblem are of finite size. Plan generating module 110 may iterativelyfind and forbid plans and may exploit the new reformulation. Plangenerating module 110 may provide a reformulation that forbids plansthat are supersets as sets rather than multisets, depicted by top-SQ.This stricter reformulation top- SQ, can be preferred if it is importantto forbid alternative but similar ways of achieving subgoals. Plangenerating module 110 may generate subset top-quality solutions indomains with infinite size unordered top-quality solutions. While eachof the approaches to top-quality planning has its own motivatingscenario, the two new ones show a great promise, particularly in domainswith redundant and zero-cost operators. Plan generating module 110 mayprove that these solutions can be obtained by exploiting the proposedreformulations. The solution sizes significantly decrease, making thenew approach more practical, particularly in domains with redundantoperators. Plan generating module 110 may implement a diverse planner ontop of a forbid iterative planner collection, on top of a fast downwardplanning system. Plan generating module 110 may improving theperformance of various planners, possibly by reusing information fromone iteration for the next one.

Further, in the depicted embodiment, plan generating module 110 includesreformulation module 112. In the depicted embodiment, reformulationmodule 112 is located on plan generating module 110 and computing device102. However, in other embodiments, reformulation module 112 may belocated externally and accessed through a communication network such asnetwork 108.

In one or more embodiments, reformulation module 112 is configured toreformulate a planning problem into a new planning problem to forbidplans that correspond to a super set of the plan. Reformulation module112 may find a plan, forbid the plan by reformulating the problem andfind the next plan. Reformulation module 112 may forbid exactly a planprovided and all its possible super-multisets. Reformulation module 112may forbid exactly a collection of plans provided and all their possiblesuper-multisets. Reformulation module 112 may reformulate a finitedomain representation planning problem into a new finite domainrepresentation planning problem so that exactly the found plans and allplans that correspond to super-sets of those plans are forbidden in aniteration. For example, reformulation module 112 may extend variables totrack operators on the previously found plans and operator sets checked.Reformulation module 112 may modify and extend the operators to changethe values of extra variables. Reformulation module 112 may extend aninitial state to allocate an initial value to extra variables.Reformulation module 112 may extend a goal to allocate goal value to theoperator sets checking variable. Reformulation module 112 may forbid allthe found plans and all plans that correspond to super-multisets ofthose plans. Reformulation module 112 may compute a set of top qualityor diverse plans, extending the criteria of equivalence fromset/multiset equivalence to super-(multi)sets. Reformulation module 112may find a plan, forbid the plan by reformulating the problem, and findthe next plan. Reformulation module 112 may forbid exactly a planprovided and all the plan's possible super-multisets. Reformulationmodule 112 may forbid exactly a collection of plans provided and alltheir possible super-multisets.

FIG. 2 is a flowchart 200 depicting operational steps of plan generatingmodule 110 in accordance with an embodiment of the present disclosure.

Plan generating module 110 operates to formulate an artificialintelligence planning problem. Plan generating module 110 may receivethe artificial intelligence planning problem for various applications,e.g., a risk management application, an energy domain application, ahealthcare application, and a malware detection application. Plangenerating module 110 also operates to receive a pre-defined stoppingcriterion for generating one or more plans for the artificialintelligence planning problem. The pre-defined stopping criterion may beassociated with one or more of a number of artificial intelligence plansneeded or a quality of artificial intelligence plans needed. Plangenerating module 110 operates to generate one or more plans byexecuting a planning algorithm (e.g., an artificial intelligenceplanner). Plan generating module 110 operates to reformulate anartificial intelligence planning problem into a new artificialintelligence planning problem to forbid plans that correspond to a superset of the plan. Plan generating module 110 operates to generates one ormore new plans based on the reformulation until the pre-defined stoppingcriterion is reached.

In step 202, plan generating module 110 formulates an artificialintelligence planning problem. Plan generating module 110 may receivethe artificial intelligence planning problem for various applications,e.g., a risk management application, an energy domain application, ahealthcare application, and a malware detection application. Plangenerating module 110 may compute a set of top-quality plans viaartificial intelligence planning algorithms. For example, plangenerating module 110 may find a strategy of action, provided that theplanning problem is modeled in a suitable input language. Plangenerating module 110 may find the one best solution to a problem, forexample, in a top K planning seeking to find the K best solutions. Forexample, given quality bound q, the top-quality planning is to findplans with quality under the bound. An unordered top-quality planning isto skip plans that are (multi)set-equivalent to previously found ones. Adiverse planning is to find k plans of a particular requirement ondiversity (e.g., variety of computational problems), given a bound k onthe number of plans and a diversity metric D. In an example, plangenerating module 110 may consider two plans are related if one'soperator multiset is a subset of the other's. Plan generating module 110may provide reformulations that forbid plans that are related to thegiven ones. While the new relation may not be transitive and thus not anequivalence relation, plan generating module 110 may define a new subsettop-quality planning problem, with finite size solution sets.

In step 204, plan generating module 110 receives a pre-defined stoppingcriterion for generating one or more plans for the artificialintelligence planning problem. The pre-defined stopping criterion may beassociated with one or more of a number of artificial intelligence plansneeded or a quality of artificial intelligence plans needed. Forexample, the pre-defined stopping criterion can be a stopping criterionof either a quality bound or a number of plans or both. Top-qualityplanning may deal with generating all high-quality plans up to a certainbound. The need for producing such a collection of plans may beestablished for many applications, including plan recognition, businessprocess automation, and automated machine learning. Top-quality planningmay serve as a basis for solving other computational problems, such asquality-aware diverse planning. In these and other applications, thechoices of planning models, whether avoidable or not, may haveunintended effects on plans. These include (zero-cost) loops,unnecessary repeated operator applications, continuing after the goal isreached. While producing all high-quality plans can be challenging, andeven impossible when there are infinitely many such plans, one practicalapproach suggested to generate all plans that are not equivalent whenignoring operator ordering.

In step 206, plan generating module 110 generates one or more plans byexecuting a planning algorithm (e.g., an artificial intelligenceplanner). In an example, a planning algorithm can be a cost-optimalplanner, an agile planner, a satisficing planner, a top-k planner, atop-quality planner, a diverse planner or any other suitable planner.Plan generating module 110 may obtain a plan by executing the planningalgorithm. Plan generating module 110 may utilize a cost-optimalartificial intelligence planner to generate an updated solution to anartificial intelligence planning problem based on a pre-defined stoppingcriterion. Plan generating module 110 may run a cost-optimal artificialintelligence planner to obtain one plan. In another example, plangenerating module 110 may run a top-k planner, a top-quality planner, adiverse planner, an agile planner, a satisficing planner, or any otherplanner to generate a solution to a planning problem. Plan generatingmodule 110 may compute a set of top-quality plans using a post-processplan by filtering out equivalent or irrelevant plans. Plan generatingmodule 110 may translate the remaining plans into a solution for anapplication. Plan generating module 110 may compute a set of top qualityor diverse plans, extending the criteria of equivalence fromset/multiset equivalence to super-(multi)sets.

In step 208, plan generating module 110 reformulates an artificialintelligence planning problem into a new artificial intelligenceplanning problem to forbid plans that correspond to a super set of theplan. Plan generating module 110 may reformulate a finite domainrepresentation planning problem into a new finite domain representationplanning problem so that exactly the found plans and all plans thatcorrespond to super-sets of those plans are forbidden in an iteration.For example, plan generating module 110 may extend variables to trackoperators on the previously found plans and operator sets checked. Plangenerating module 110 may modify and extend the operators to change thevalues of extra variables. Plan generating module 110 may extend aninitial state to allocate an initial value to extra variables. Plangenerating module 110 may extend a goal to allocate goal value to theoperator sets checking variable. Plan generating module 110 may forbidall the found plans and all plans that correspond to super-multisets ofthose plans. Plan generating module 110 may compute a set of top qualityor diverse plans, extending the criteria of equivalence fromset/multiset equivalence to super-(multi)sets. Plan generating module110 may find a plan, forbid the plan by reformulating the problem, andfind the next plan. Plan generating module 110 may forbid exactly a planprovided and all the plan's possible super-multisets. Plan generatingmodule 110 may forbidding exactly a collection of plans provided and alltheir possible super-multisets.

In step 210, plan generating module 110 generates one or more new plansbased on the reformulation until the pre-defined stopping criterion isreached. Plan generating module 110 may run until a stopping criterionis reached: either a plan of worse quality than the quality bound isfound, or enough plans were found, or there are no more plans exist. Inan example, plan generating module 110 may generate a subset top-qualitysolution in domain with infinite size unordered top-quality solutions.Plan generating module 110 may utilize a cost-optimal planner togenerate an updated solution to the artificial intelligence planningproblem based on the pre-defined stopping criterion. Plan generatingmodule 110 may solve a computational problem under the umbrella oftop-quality planning that allows to further restrict the solution set.Plan generating module 110 may consider the following relation: a planis related to another if its operator multiset is a subset of theother's. Plan generating module 110 may provide a planning taskreformulation that forbids plans whose operator multisets are supersetsof the given ones. Plan generating module 110 may prove that thesolution sets of the new subset top-quality planning problem are offinite size. Plan generating module 110 may iteratively find and forbidplans and may exploit the new reformulation. Plan generating module 110may provide a reformulation that forbids plans that are supersets assets rather than multisets, depicted by top-SQ. This stricterreformulation top-SQ, can be preferred if it is important to forbidalternative but similar ways of achieving subgoals. Plan generatingmodule 110 may generate subset top-quality solutions in domains withinfinite size unordered top- quality solutions. While each of theapproaches to top-quality planning has its own motivating scenario, thetwo new ones show a great promise, particularly in domains withredundant and zero-cost operators. Plan generating module 110 may provethat these solutions can be obtained by exploiting the proposedreformulations. The solution sizes significantly decrease, making thenew approach more practical, particularly in domains with redundantoperators.

Plan generating module 110 may implement a diverse planner on top of aforbid iterative planner collection, on top of a fast downward planningsystem. Plan generating module 110 may improving the performance ofvarious planners, possibly by reusing information from one iteration forthe next one.

FIG. 3 illustrates an exemplary functional diagram of plan generatingmodule 110 in automating artificial intelligence pipeline generationenvironment 100, in accordance with an embodiment of the presentdisclosure.

In the example of FIG. 3 , inputs 326 includes training and test data302, data science grammar 304, domain specific constraints 306 andplanner configuration 308. Outputs 328 includes trained pipelines 322and trainable pipelines 324. Automating artificial intelligence pipelinegeneration environment 100 also includes grammar to HTN (HierarchicalTask Network) translator 310, HTN to PDDL (Planning Domain DefinitionLanguage) 312, PDDL updates 314, PDDL planner 316 (e.g., plan generatingmodule 110), optimizer 318 and plan to machine learning pipeline 320.

FIG. 4 illustrates exemplary operational steps of plan generating module110 in accordance with an embodiment of the present disclosure.

In the example of FIG. 4 , at block 402, plan generating module 110receives a planning problem and stopping criteria as an input. At block404, plan generating module 110 uses a state of-the-art cost-optimalplanner to find optimal plan(s). Plan generating module 110 may find oneplan or a set of plans. At block 406, plan generating module 110reformulates the planning problem to forbid plans. Plan generatingmodule 110 may forbid this set of plans and those that correspond tosupersets of the operators in the plans. At block 408, plan generatingmodule 110 stops running until the stopping criteria is reached, e.g.,either a plan of worse quality than the quality bound is found, orenough plans were found, or there are no more plans exist.

FIG. 5 illustrates an exemplary experimental evaluation of plangenerating module 110 about per-domain coverage for configurations, inaccordance with an embodiment of the present disclosure.

In the example of FIG. 5 , to empirically evaluate the feasibility ofthe approach, plan generating module 110 may implement diverse plannerson top of a forbid iterative planner collection and a fast downwardplanning system. The two example reformulations depicted in Definition 2(denoted by top-MSQ 506) and Definition 3 (denoted by top-SQ 508) arecompared to the unordered top-quality planning (top-uQ 504). Theevaluation may focus on cost-optimal plans, e.g., q=1. FIG. 5 depictsthe per-domain coverage of the three approaches, with the largestcoverage bolded. A planner gets coverage of 1 for a task if it was ableto find the solution for its corresponding computational problem. Fortop-uQ 504 it means finding all cost-optimal plans up to reordering. Fortop-MSQ 506 (top-SQ 508) it means finding all cost-optimal plans up tosubmultiple-sets (subsets). While on most of the 64 domains 502 there isno change in the coverage, there are 16 domains depicted in FIG. 5 wherethe coverage differs. Note that the overall coverage significantlyincreases compared to the baseline, for both approaches. There are fourdomains 502 where the previous approach could not solve any task, butthe current proposed approach can. Out of these, elevators and Sokobancould not be solved by the previous approach in principle, as theunordered top- quality solution has an infinite number of plans. Thesuperset top-quality one is however finite in these domains. Still, in afew domains (e.g., movie, openstacks, parcprinter, pipesworld, storage)the baseline top-uQ still dominates. The corresponding reformulation issimpler than the suggested ones, and therefore the classical plannersare expected to be able to solve these reformulated tasks quicker. Ifthe number of iterations does not (significantly) decrease whenattempting to forbid more plans, the baseline should perform better thanthe proposed approaches. These new suggested approaches thrive when thesolution size decreases significantly.

FIG. 6 illustrates an exemplary experimental evaluation of plangenerating module 110, in accordance with an embodiment of the presentdisclosure.

In the example of FIG. 6 , the number of plans in the solution for thethree approaches are compared to verify that the solution sizes indeeddecrease. Out of 327 tasks solved by all three approaches, on 277 theplan sets are of the same size. The remaining 50 tasks are depicted inFIG. 6 , comparing top-SQ 602 to top-uQ 604. Each point in the plotcorresponds to a planning task that was solved by both approaches, anddepicting the number of plans in each solution. The 50 tasks depicted inFIG. 6 all belong to the following four domains: ged, pegsol, psr, andspider. For these tasks, the solution size is decreased by half onaverage.

FIG. 7 depicts a block diagram 700 of components of computing device 102in accordance with an illustrative embodiment of the present disclosure.It should be appreciated that FIG. 7 provides only an illustration ofone implementation and does not imply any limitations with regard to theenvironments in which different embodiments may be implemented. Manymodifications to the depicted environment may be made.

Computing device 102 may include communications fabric 702, whichprovides communications between cache 716, memory 706, persistentstorage 708, communications unit 710, and input/output (I/O)interface(s) 712. Communications fabric 702 can be implemented with anyarchitecture designed for passing data and/or control informationbetween processors (such as microprocessors, communications and networkprocessors, etc.), system memory, peripheral devices, and any otherhardware components within a system. For example, communications fabric702 can be implemented with one or more buses or a crossbar switch.

Memory 706 and persistent storage 708 are computer readable storagemedia. In this embodiment, memory 706 includes random access memory(RAM). In general, memory 706 can include any suitable volatile ornon-volatile computer readable storage media. Cache 716 is a fast memorythat enhances the performance of computer processor(s) 704 by holdingrecently accessed data, and data near accessed data, from memory 706.

Plan generating module 110 may be stored in persistent storage 708 andin memory 706 for execution by one or more of the respective computerprocessors 704 via cache 716. In an embodiment, persistent storage 708includes a magnetic hard disk drive. Alternatively, or in addition to amagnetic hard disk drive, persistent storage 708 can include a solidstate hard drive, a semiconductor storage device, read-only memory(ROM), erasable programmable read-only memory (EPROM), flash memory, orany other computer readable storage media that is capable of storingprogram instructions or digital information.

The media used by persistent storage 708 may also be removable. Forexample, a removable hard drive may be used for persistent storage 708.Other examples include optical and magnetic disks, thumb drives, andsmart cards that are inserted into a drive for transfer onto anothercomputer readable storage medium that is also part of persistent storage708.

Communications unit 710, in these examples, provides for communicationswith other data processing systems or devices. In these examples,communications unit 710 includes one or more network interface cards.Communications unit 710 may provide communications through the use ofeither or both physical and wireless communications links. Plangenerating module 110 may be downloaded to persistent storage 708through communications unit 710.

I/O interface(s) 712 allows for input and output of data with otherdevices that may be connected to computing device 102. For example, I/Ointerface 712 may provide a connection to external devices 718 such as akeyboard, keypad, a touch screen, and/or some other suitable inputdevice. External devices 718 can also include portable computer readablestorage media such as, for example, thumb drives, portable optical ormagnetic disks, and memory cards. Software and data used to practiceembodiments of the present invention, e.g., plan generating module 110can be stored on such portable computer readable storage media and canbe loaded onto persistent storage 708 via I/O interface(s) 712. I/Ointerface(s) 712 also connect to display 720.

Display 720 provides a mechanism to display data to a user and may be,for example, a computer monitor.

The programs described herein are identified based upon the applicationfor which they are implemented in a specific embodiment of theinvention. However, it should be appreciated that any particular programnomenclature herein is used merely for convenience, and thus theinvention should not be limited to use solely in any specificapplication identified and/or implied by such nomenclature.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions, state-setting data, configuration data for integrated circuitry, or eithersource code or object code written in any combination of one or moreprogramming languages, including an object oriented programming languagesuch as Python, C++, or the like, and procedural programming languages,such as the “C” programming language or similar programming languages.The computer readable program instructions may execute entirely on theuser's computer, partly on the user's computer, as a stand-alonesoftware package, partly on the user's computer and partly on a remotecomputer or entirely on the remote computer or server. In the latterscenario, the remote computer may be connected to the user's computerthrough any type of network, including a local area network (LAN) or awide area network (WAN), or the connection may be made to an externalcomputer (for example, through the Internet using an Internet ServiceProvider). In some embodiments, electronic circuitry including, forexample, programmable logic circuitry, field-programmable gate arrays(FPGA), or programmable logic arrays (PLA) may execute the computerreadable program instructions by utilizing state information of thecomputer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a computer, or other programmable data processing apparatusto produce a machine, such that the instructions, which execute via theprocessor of the computer or other programmable data processingapparatus, create means for implementing the functions/acts specified inthe flowchart and/or block diagram block or blocks. These computerreadable program instructions may also be stored in a computer readablestorage medium that can direct a computer, a programmable dataprocessing apparatus, and/or other devices to function in a particularmanner, such that the computer readable storage medium havinginstructions stored therein comprises an article of manufactureincluding instructions which implement aspects of the function/actspecified in the flowchart and/or block diagram block or blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be accomplished as one step, executed concurrently,substantially concurrently, in a partially or wholly temporallyoverlapping manner, or the blocks may sometimes be executed in thereverse order, depending upon the functionality involved. It will alsobe noted that each block of the block diagrams and/or flowchartillustration, and combinations of blocks in the block diagrams and/orflowchart illustration, can be implemented by special purposehardware-based systems that perform the specified functions or acts orcarry out combinations of special purpose hardware and computerinstructions.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the invention.The terminology used herein was chosen to best explain the principles ofthe embodiment, the practical application or technical improvement overtechnologies found in the marketplace, or to enable others of ordinaryskill in the art to understand the embodiments disclosed herein.

Although specific embodiments of the present invention have beendescribed, it will be understood by those of skill in the art that thereare other embodiments that are equivalent to the described embodiments.Accordingly, it is to be understood that the invention is not to belimited by the specific illustrated embodiments, but only by the scopeof the appended claims.

What is claimed is:
 1. A computer-implemented method comprising:formulating, by one or more processors, an artificial intelligenceplanning problem; receiving, by one or more processors, a pre-definedstopping criterion for generating one or more plans for the artificialintelligence planning problem; generating, by one or more processors,the one or more plans by executing a planning algorithm; reformulating,by one or more processors, the artificial intelligence planning probleminto a new artificial intelligence planning problem by forbidding plansthat correspond to super-sets of the one or more plans; and generating,by one or more processors, one or more new plans based on thereformulation until the pre-defined stopping criterion is reached. 2.The computer-implemented method of claim 1, wherein reformulating theartificial intelligence planning problem comprises: extending to trackone or more operators on a previously found plan; modifying andextending the one or more operator to change a value of an extravariable; extending an initial state to allocate an initial value to theextra variable; and extending a goal to allocate a goal value to the oneor more operators and to check the extra variable.
 3. Thecomputer-implemented method of claim 1, wherein reformulating theartificial intelligence planning problem comprises: forbidding plansthat correspond to super-multisets of the one or more plans.
 4. Thecomputer-implemented method of claim 1, wherein the pre-defined stoppingcriterion is selected from a group consisting of: a plan of worsequality than the quality bound is found, enough plans are found, and nomore plans exist.
 5. The computer-implemented method of claim 1, whereinthe planning algorithm is selected from a group consisting of: acost-optimal planner, an agile planner, a satisficing planner, a top-kplanner, a top-quality planner, and a diverse planner.
 6. Thecomputer-implemented method of claim 1, further comprising: generating asubset top-quality solution in domain with infinite size unordered top-quality solutions.
 7. The computer-implemented method of claim 1,further comprising: utilizing a cost-optimal planner to generate anupdated solution to the artificial intelligence planning problem basedon the pre-defined stopping criterion.
 8. A computer program productcomprising: one or more computer readable storage media, and programinstructions collectively stored on the one or more computer readablestorage media, the program instructions comprising: program instructionsto formulate an artificial intelligence planning problem; programinstructions to receive a pre-defined stopping criterion for generatingone or more plans for the artificial intelligence planning problem;program instructions to generate the one or more plans by executing aplanning algorithm; program instructions to reformulate the artificialintelligence planning problem into a new artificial intelligenceplanning problem by forbidding plans that correspond to super-sets ofthe one or more plans; and program instructions to generate one or morenew plans based on the reformulation until the pre-defined stoppingcriterion is reached.
 9. The computer program product of claim 8,wherein program instructions to reformulate the artificial intelligenceplanning problem comprise: program instructions to extend to track oneor more operators on a previously found plan; program instructions tomodify and extend the one or more operator to change a value of an extravariable; program instructions to extend an initial state to allocate aninitial value to the extra variable; and program instructions to extenda goal to allocate a goal value to the one or more operators and tocheck the extra variable.
 10. The computer program product of claim 8,wherein program instructions to reformulate the artificial intelligenceplanning problem comprises: program instructions to forbid plans thatcorrespond to super-multisets of the one or more plans.
 11. The computerprogram product of claim 8, wherein the pre-defined stopping criterionis selected from a group consisting of: a plan of worse quality than thequality bound is found, enough plans are found, and no more plans exist.12. The computer program product of claim 8, wherein the planningalgorithm is selected from a group consisting of: a cost-optimalplanner, an agile planner, a satisficing planner, a top-k planner, atop-quality planner, and a diverse planner.
 13. The computer programproduct of claim 8, further comprising: program instructions to generatea subset top-quality solution in domain with infinite size unorderedtop-quality solutions.
 14. The computer program product of claim 8,further comprising: program instructions to utilize a cost-optimalplanner to generate an updated solution to the artificial intelligenceplanning problem based on the pre-defined stopping criterion.
 15. Acomputer system comprising: one or more computer processors, one or morecomputer readable storage media, and program instructions stored on theone or more computer readable storage media for execution by at leastone of the one or more computer processors, the program instructionscomprising: program instructions to formulate an artificial intelligenceplanning problem; program instructions to receive a pre-defined stoppingcriterion for generating one or more plans for the artificialintelligence planning problem; program instructions to generate the oneor more plans by executing a planning algorithm; program instructions toreformulate the artificial intelligence planning problem into a newartificial intelligence planning problem by forbidding plans thatcorrespond to super-sets of the one or more plans; and programinstructions to generate one or more new plans based on thereformulation until the pre-defined stopping criterion is reached. 16.The computer system of claim 15, wherein program instructions toreformulate the artificial intelligence planning problem comprise:program instructions to extend to track one or more operators on apreviously found plan; program instructions to modify and extend the oneor more operator to change a value of an extra variable; programinstructions to extend an initial state to allocate an initial value tothe extra variable; and program instructions to extend a goal toallocate a goal value to the one or more operators and to check theextra variable.
 17. The computer system of claim 15, wherein programinstructions to reformulate the artificial intelligence planning problemcomprises: program instructions to forbid plans that correspond tosuper-multisets of the one or more plans.
 18. The computer system ofclaim 15, wherein the pre-defined stopping criterion is selected from agroup consisting of: a plan of worse quality than the quality bound isfound, enough plans are found, and no more plans exist.
 19. The computersystem of claim 15, wherein the planning algorithm is selected from agroup consisting of: a cost-optimal planner, an agile planner, asatisficing planner, a top-k planner, a top-quality planner, and adiverse planner.
 20. The computer system of claim 15, furthercomprising: program instructions to generate a subset top-qualitysolution in domain with infinite size unordered top-quality solutions.